A segment has a midpoint (-2, 9) and one endpoint (2,8). What is the coordinate of the other endpoint?

2 Answers
Jun 7, 2017

See a solution process below:

Explanation:

The formula to find the mid-point of a line segment give the two end points is:

M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)

Where M is the midpoint and the given points are:

(color(red)(x_1, y_1)) and (color(blue)(x_2, y_2))

Subtituting gives:

(-2, 9) = ((color(red)(2) + color(blue)(x_2))/2 , (color(red)(8) + color(blue)(y_2))/2)

Solving for x gives:

-2 = (color(red)(2) + color(blue)(x_2))/2

color(green)(2) xx -2 = color(green)(2) xx (color(red)(2) + color(blue)(x_2))/2

-4 = cancel(color(green)(2)) xx (color(red)(2) + color(blue)(x_2))/color(green)(cancel(color(black)(2)))

-4 = color(red)(2) + color(blue)(x_2)

-2 - 4 = -2 + color(red)(2) + color(blue)(x_2)

-6 = 0 + color(blue)(x_2)

-6 = color(blue)(x_2)

color(blue)(x_2) = -6

Solving for y gives:

9 = (color(red)(8) + color(blue)(y_2))/2

color(green)(2) xx 9 = color(green)(2) xx (color(red)(8) + color(blue)(y_2))/2

18 = cancel(color(green)(2)) xx (color(red)(8) + color(blue)(y_2))/color(green)(cancel(color(black)(2)))

18 = color(red)(8) + color(blue)(y_2)

-8 + 18 = -8 + color(red)(8) + color(blue)(y_2)

10 = 0 + color(blue)(y_2)

10 = color(blue)(y_2)

color(blue)(y_2) = 10

The other end point is: (color(blue)(-6, 10))

Jun 7, 2017

The coordinate of the other endpoint is (-6, 10)

Explanation:

Instead of full-on visualizing a graph, let's start with a number line:

This is a simple number line from 0 to 10. Now, I ask you, what is an easy way to calculate the midpoint between 0 and 10? Of course it's 5 but is there a formula we can derive to calculate the midpoint between any two points? Well, let's see. If we took 0 and 10, added them together, then divided the quotient by 2, boom! We get 5!

Let's try it with two more numbers. How about 2 and 4? If we were to add the two up, then divide the result by 2, we'd get 3 - also the midpoint between 2 and 4.

Now we can see that there's a formula that we can use to calculate the midpoint between any two numbers! Add the two end numbers up and divide the result by 2! This formula - known as the Midpoint Formula - is shown below:

For any two endpoints x_1 and x_2 on the number line,

M=(x_1+x_2)/2

So how does all of this relate to the problem? Well, remember that the graph is a coordinate plane, made up of two axes - the x-axis and the y-axis. And you can think of each axis to be a number line!

So now, our task is to derive a formula for finding the midpoint between any two points on the coordinate point. That way, we can write a relationship between the two endpoints on the coordinate plane and the midpoint between those two points.

Suppose there was a graph like the one shown below:

Suppose point A had the coordinates (x_1, y_1) and point B had the coordinates (x_2, y_2). And suppose there was also a point M that was the midpoint between points A and B. Now, we want to write the midpoint's coordinates in terms of the x and y coordinates of points A and B. That way, we can connect the coordinates of points A and B with the coordinates of their midpoint. Now, the coordinates of the midpoint are essentially the midpoint between the two x-coordinates and the midpoint between the two y-coordinates.

Let's focus on the x-axis first and treat it as a number line, except without numbers. On the x-axis, we have the x-coordinate of point A as well as the x-coordinate of point B. From our earlier rule, we can state that the midpoint between the two x-coordinates is:

M=(x_1+x_2)/2

Likewise, the midpoint between the two y-coordinates is:

M=(y_1+y_2)/2

As we stated earlier, the coordinates of the midpoint between points A and B are the midpoint between the two x-coordinates and the midpoint between the two y-coordinates. Combining the two statements above, we get the conclusion that the coordinates of the midpoint between points A and B are:

M=((x_1+x_2)/2, (y_1+y_2)/2)

Now, with that information, we can substitute in the values mentioned in the question. We let point A and its coordinates be our missing endpoint, point B and its coordinates be our known endpoint, and point M, our midpoint, and its coordinates be the midpoint, like so:

M(-2, 9)=((x_1+2)/2, (y_1+8)/2)

This equation tells us that -2=(x_1+2)/2 and that 9=(y_1+8)/2. Next, we start solving the equations to figure out what x_1 and y_1 are.

Let's start with the left equation:

-2=(x_1+2)/2

x_1+2=-4

x_1=-6

And now, let's solve the right equation:

9=(y_1+8)/2

y_1+8=18

y_1=10

And now, we put the two values together to form the answer:

(-6, 10)